Let O (0,0),P(3,4),Q(6,0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR,PQR,OQR are of equal area. The coordinates of R are

Options

A. `(3,2/3)`
B. `(4/3,3)`
C. `(3,4/3)`
D. `(4/3,2/3)`

Answer

To find the coordinates of point R inside triangle OPQ such that the areas of triangles OPR, PQR, and OQR are equal, we can use the property of the centroid of a triangle. The centroid divides the triangle into three smaller triangles of equal area.

### Step-by-Step Solution:

1. **Identify the vertices of the triangle OPQ:**
- O(0, 0)
- P(3, 4)
- Q(6, 0)

2. **Calculate the coordinates of the centroid (R):**
The formula for the centroid (G) of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

3. **Substituting the coordinates of points O, P, and Q into the centroid formula:**
- \(x_1 = 0\), \(y_1 = 0\)
- \(x_2 = 3\), \(y_2 = 4\)
- \(x_3 = 6\), \(y_3 = 0\)

Now, calculate the x-coordinate of the centroid:
\[ x_G = \frac{0 + 3 + 6}{3} = \frac{9}{3} = 3 \]

Next, calculate the y-coordinate of the centroid:
\[ y_G = \frac{0 + 4 + 0}{3} = \frac{4}{3} \]

4. **Thus, the coordinates of point R (the centroid) are:**
\[ R\left(3, \frac{4}{3}\right) \]

### Final Answer:
The coordinates of point R are \(R\left(3, \frac{4}{3}\right)\).

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